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Archive for Mathematical Logic

, Volume 51, Issue 3–4, pp 257–283 | Cite as

Inner models with large cardinal features usually obtained by forcing

  • Arthur W. Apter
  • Victoria Gitman
  • Joel David Hamkins
Article

Abstract

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2 κ  = κ +, another for which 2 κ  = κ ++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal κ, such that \({H^{V}_{\kappa^+} \subseteq {\rm HOD}^W}\). Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH + V = HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit δ of <δ-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results.

Keywords

Forcing Large cardinals Inner models 

Mathematics Subject Classification (2000)

03E45 03E55 03E40 

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References

  1. 1.
    Apter A.W.: On level by level equivalence and inequivalence between strong compactness and supercompactness. Fund. Math. 171(1), 77–92 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Apter A.W.: Diamond, square, and level by level equivalence. Arch. Math. Log. 44(3), 387–395 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Apter A.W.: Tallness and level by level equivalence and inequivalence. Math. Log. Q. 56(1), 4–12 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Apter A.W.: Level by level inequivalence beyond measurability. Arch. Math. Log. 50(7-8), 707–712 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Apter A.W., Shelah S.: On the strong equality between supercompactness and strong compactness. Trans. Am. Math. Soc. 349(1), 103–128 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Baumgartner J.E.: Applications of the proper forcing axiom. In: Kunen, K., Vaughan, J. (eds) Handbook of Set Theoretic Topology, pp. 913–959. North-Holland, Amsterdam (1984)Google Scholar
  7. 7.
    Brooke-Taylor A.: Large cardinals and definable well-orders on the universe. J. Symb. Log. 74(2), 641–654 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dobrinen N., Friedman S.D.: Internal consistency and global co-stationarity of the ground model. J. Symb. Log. 73(2), 512–521 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dobrinen N., Friedman S.D.: The consistency strength of the tree property at the double successor of a measurable cardinal. Fund. Math. 208(2), 123–153 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Foreman M.D.: Smoke and mirrors: combinatorial properties of small cardinals equiconsistent with huge cardinals. Adv. Math. 222(2), 565–595 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Friedman S.D.: Internal consistency and the inner model hypothesis. Bull. Symb. Log. 12(4), 591–600 (2006)zbMATHCrossRefGoogle Scholar
  12. 12.
    Friedman, S.: Aspects of HOD, supercompactness, and set theoretic geology. PhD thesis, The Graduate Center of the City University of New York (2009)Google Scholar
  13. 13.
    Gitman, V., Hamkins, J.D., Johnstone, T.A.: What is the theory ZFC without power set? (submitted for publication)Google Scholar
  14. 14.
    Gitman V.: Ramsey-like cardinals. J. Symb. Log. 76(2), 519–540 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Gitman V., Welch P.D.: Ramsey-like cardinals II. J. Symb. Log. 76(2), 541–560 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Hamkins J.D.: Fragile measurability. J. Symb. Log. 59(1), 262–282 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Hamkins J.D.: The lottery preparation. Ann. Pure Appl. Log. 101, 103–146 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Hamkins J.D.: The ground axiom. Oberwolfach Rep. 55, 3160–3162 (2005)Google Scholar
  19. 19.
    Hamkins, J.D., Seabold, D.: Boolean ultrapowers (in preparation)Google Scholar
  20. 20.
    Kunen K.: Some applications of iterated ultrapowers in set theory. Ann. Math. Log. 1, 179–227 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Laver R.: Making the supercompactness of κ indestructible under κ-directed closed forcing. Israel J. Math. 29(4), 385–388 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Lévy A., Solovay R.M.: Measurable cardinals and the continuum hypothesis. Israel J. Math. 5, 234–248 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Magidor M.: How large is the first strongly compact cardinal? or a study on identity crises. Ann. Math. Log. 10(1), 33–57 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Mitchell W.J.: Ramsey cardinals and constructibility. J. Symb. Log. 44(2), 260–266 (1979)zbMATHCrossRefGoogle Scholar
  25. 25.
    Reitz, J.: The ground axiom. PhD thesis, The Graduate Center of the City University of New York (2006)Google Scholar
  26. 26.
    Reitz J.: The ground axiom. J. Symb. Log. 72(4), 1299–1317 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Solovay R.M. et al.: Strongly compact cardinals and the GCH. In: Henkin, L. (ed) Proceedings of the Tarski Symposium, Proceedings Symposia Pure Mathematics, volume XXV, University of California, Berkeley, 1971, pp. 365–372. American Mathematical Society, Providence, Rhode Island (1974)Google Scholar
  28. 28.
    Welch P.: On unfoldable cardinals, omega cardinals, and the beginning of the inner model hierarchy. Arch. Math. Log. 43(4), 443–458 (2004)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Arthur W. Apter
    • 1
    • 2
  • Victoria Gitman
    • 3
  • Joel David Hamkins
    • 1
    • 4
    • 5
  1. 1.MathematicsThe Graduate Center of the City University of New YorkNew YorkUSA
  2. 2.Department of MathematicsBaruch College of CUNYNew YorkUSA
  3. 3.MathematicsNew York City College of Technology of CUNYBrooklynUSA
  4. 4.Department of PhilosophyNew York UniversityNew YorkUSA
  5. 5.MathematicsThe College of Staten Island of CUNYStaten IslandUSA

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